GATE CS 2015 Set 1 Q.28 (CS Q.18): Linear Algebra

In the LU decomposition of the matrix $\begin{bmatrix} 2 & 2 \\ 4 & 9 \end{bmatrix}$ , if the diagonal elements of $U$ are both $1$ , then the lower diagonal entry $l_{22}$ of $L$ is _____.

Your answer

Official answer and solution

Official answer: 5
From: Official GATE 2015 answer key

Solution

Factor matrix A into L and U with unit diagonal in U (Crout's LU decomposition).

Let $A = \begin{bmatrix} 2 & 2 \\ 4 & 9 \end{bmatrix}$ .
We are given that $A = L U$ where $L$ is lower triangular and $U$ is upper triangular with diagonal elements equal to $1$ :

L = \begin{bmatrix} l_{11} & 0 \\ l_{21} & l_{22} \end{bmatrix}, \quad U = \begin{bmatrix} 1 & u_{12} \\ 0 & 1 \end{bmatrix}.

Multiplying $L$ and $U$ :

LU = \begin{bmatrix} l_{11} & l_{11} u_{12} \\ l_{21} & l_{21} u_{12} + l_{22} \end{bmatrix} = \begin{bmatrix} 2 & 2 \\ 4 & 9 \end{bmatrix}.

Equating matrix components step-by-step:

First column:
$l_{11} = 2$ $l_{21} = 4$
First row, second column:
$l_{11} u_{12} = 2 \implies 2 u_{12} = 2 \implies u_{12} = 1$
Second row, second column:
$l_{21} u_{12} + l_{22} = 9$ $4(1) + l_{22} = 9 \implies l_{22} = 9 - 4 = 5.$

The lower diagonal entry $l_{22}$ of $L$ is $5$ .

Final answer

Common trap

Assuming Doolittle's decomposition where L has unit diagonal ( $l_{ii} = 1$ ) rather than U as explicitly specified.

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2015 Computer Science and Information Technology (CS) Session 1
Question: 28 of 65, printed as CS Q.18
Section: Computer Science and Information Technology

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